The mathematical derivative in its classic origin story comes from setting something paradoxically to zero. But it can't actually be zero, because then our math

The mathematical derivative in its classic origin story comes from setting something paradoxically to zero. But it can't actually be zero, because then our math breaks: "the lingering essence of a departed quantity". The derivative is seen in the real world in things like motion. If we know something's position changes over time we can calculate its speed (over time), and acceleration (the derivative of speed over time). Motion is (kinetic) energy. Energy, manifested as motion, is transferred to an object not as force over time ('tis but a proxy) but as force over distance. This is why the continuous application of a fixed force to an object in motion transfers more energy to the object in a unit of time when the object is moving faster to begin with. (kinetic energy for an object of a certain mass is proportional to speed^2) But imagine that an object is truly "at rest". No motion. A force is applied. Why does it start moving in the first place?